Dirac systems with locally square-integrable potentials: direct and inverse problems for the spectral functions

2023/06/22 by Alexander Sakhnovich, Sakhnovich, Alexander
Mathematics · Physics and Astronomy · #34A55 #34L40 #47B15 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2306.12976

openalex publication_date 2023/06/22 · openalex created_date 2023/06/24 · openalex updated_date 2026/08/01

Abstract

We solve the inverse problems to recover Dirac systems on an interval or semiaxis from their spectral functions (matrix valued functions) for the case of locally square-integrable potentials. Direct problems in terms of spectral functions are treated as well. Moreover, we present necessary and sufficient conditions on the given distribution matrix valued function to be a spectral function of some Dirac system with a locally square-integrable potential. Interesting connections with Paley-Wiener sampling measures appear in the case of scalar spectral functions.

Related

• The inverse approach to Dirac-type systems based on the A-function\n concept
• Direct spectral problem for a Dirac operator with non-local potential
• Spectral analysis of the Dirac system with a singularity in an interior point
• Recovery of the matrix potential of the one-dimensional Dirac equation from spectral data
• Asymptotic behavior of solutions of the Dirac system with an integrable\n potential